\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
Let $D$ be a division ring with center $F$ and denote by $[D,D]$
the group generated additively by additive commutators. First, it
is shown that in zero characteristic, $D$ is algebraic over $F$ if
and only if each element of $[D,D]$ is algebraic over $F$. We
conjecture this assertion is true for any characteristic. Also, as
a generalization of Jacobson's Theorem it is proved that $D$ is an
$F$-central division ring if and only if all its additive
commutators are of bounded degree over $F$. Furthermore, we study
the $F$-vector space $D/[D,D]$ and show that $dim_F D/ [D,D] \leq
1$ if $D$ is algebraic over $F$ and $char F=0$. We then prove that
any algebraic division ring contains a separable additive
commutator over $F$ except in one special case. Finally, the
existence of primitive elements in $[D,D]$ is studied for finite
separable extensions of $F$ in $D$.
\end{document}